Reviewer:a e b
September 8, 2011 Subject:
then and now in continuous groups
Campbell's Lie Algebras and Continuous Groups is a book I bought back in 1973. It is good that Internet Archive has made it permenant by digitizing it. When Sophus Lie did his research, the mathematical terminology was not fixed as it is today. Thus readers of Campbell's book may be a bit confused. The terminology maps as follows:
what Campbell calls a 'group' is now called a 'semi-group'; what he calls 'independent' is now 'linearly indepedent', what he calls 'unconnected' is now 'algebraically independent', what he calls 'finite group' is a semi-group with finite number of parameters generating it, what he calls 'infinite group' is a semi-group with uncountable number of generators. Campbell's book is worthy of reading because of the wide range of his analysis. Yes, the treatment does look at problems of interest in late 19th century mathematics, but it is a solid treatment. There is a lot of meat in this book. Those wanting a more modern treatment of Lie algebras may consult Sagle and Walde, Introduction to Lie Groups and Lie Algebras, 1973. Those wanting an advanced treatment of continuous groups should use the classic Hille and Phillips, Functional Analysis and Semi-groups, 1957.